Could be as well typo, "29,000 feet" would make more sense (that's approximate height of Mt Everest)
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vartecSep 7 '12 at 12:25

@vartec Probably a mountain could be a bit higher. Maua Kea is famous as having bigger relative height. As guest stated, Olympus Mons also is heigher even taking into account lower Mars gravity.
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BartekChomJan 16 at 9:43

2 Answers
2

How exactly the different intrinsic and extrinsic factors interplay to shape real mountains is an active field of research. Thus, it's not possible to say exactly how high a mountain could become on earth. However, there are several limits to that.

First, there is the issue of rock stability itself. Rock has a limited compressive strength, but quite a bit of weight (relative rock density is on the order of 2.5), so if a mountain becomes too high, the rock at the base will simply crumble or melt from the pressure.

Terzagi (1962); Géotechnique, Volume 12, Issue 4, pages 251 –270 calculated the theoretical height of the tallest vertical cliff as H=strength/weight[N/m^3], which comes out to about 7.5 km for granite. Of course, a mountain is not a vertical cliff, and when you double the Granite value, you get the about 15km in the OP (full disclaimer: I'm not 100% sure how exactly adding slopes on the side gets you a factor of 2, but I'm running out of time here). Note that the above formula takes into account the weight of the rock, which means that smaller planets can have higher mountains.

The interaction between tectonism and erosion produces rugged landscapes in actively deforming regions. In the northwestern Himalaya, the form of the landscape was found to be largely independent of exhumation rates, but regional trends in mean and modal elevations, hypsometry (frequency distribution of altitude), and slope distributions were correlated with the extent of glaciation. These observations imply that in mountain belts that intersect the snowline, glacial and periglacial processes place an upper limit on altitude, relief, and the development of topography irrespective of the rate of tectonic processes operating.

The height of mountain ranges reflects the balance between tectonic rock uplift, crustal strength and surface denudation. Tectonic deformation and surface denudation are interdependent, however, and feedback mechanisms—in particular, the potential link to climate—are subjects of intense debate(1, 2). Spatial variations in fluvial denudation rate caused by precipitation gradients are known to provide first-order controls on mountain range width, crustal deformation rates and rock uplift(3, 4). Moreover, limits to crustal strength(5) are thought to constrain the maximum elevation of large continental plateaus, such as those in Tibet and the central Andes. There are indications that the general height of mountain ranges is also directly influenced by the extent of glaciation through an efficient denudation mechanism known as the glacial buzzsaw(6, 7, 8, 9). Here we use a global analysis of topography and show that variations in maximum mountain height correlate closely with climate-controlled gradients in snowline altitude for many high mountain ranges across orogenic ages and tectonic styles. With the aid of a numerical model, we further demonstrate how a combination of erosional destruction of topography above the snowline by glacier-sliding and commensurate isostatic landscape uplift caused by erosional unloading can explain observations of maximum mountain height by driving elevations towards an altitude window just below the snowline. The model thereby self-consistently produces the hypsometric signature of the glacial buzzsaw, and suggests that differences in the height of mountain ranges mainly reflect variations in local climate rather than tectonic forces.

Here's the link to ref#5, which doesn't unfortunately, calculate the maximum theoretical height of a mountain. I guess geologists may mention these things in talks, but not in high-end journal publications.

In summary: The 15km limit may be plausible, but it's unlikely to ever be attained by real-earth mountains, even the 10km ones that hide from most of erosion in the sea.

Sorry, your explanation for rock stability doesn't hold as it cannot explain how Olympus Mons on Mars can still be standing. It is 22km, or 72,000 feet tall.
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user8483Sep 6 '12 at 19:46

9

@Maxim How doesn't it? Mars has lower surface gravity and the rock experiences less compression.
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William GrobmanSep 6 '12 at 20:18

9

@Maxim - I quote: "Note that the above formula takes into account the weight of the rock, which means that smaller planets can have higher mountains." Last I checked, Mars was still less massive than the Earth, although we humans have begun using it as a dump site for our discarded rovers. I won't get into the differences in glaciation, since we are doing our best to melt the glaciers anyway.
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user3344Sep 6 '12 at 20:29

An engineer and instructor from NASA posting on Quora had this to say about maximal height of mountains on Earth:

Gravity has a direct inverse proportional relationship. So a planet
with a gravitational acceleration twice that of Earth could have a
mountain half as tall as one on Earth and a planet with a
gravitational acceleration half that of Earth could have a mountain
twice as tall as one on Earth.

There are three more significant constraints and they all relate to
the geology:

1) What is the mountain made of? Put enough rocks on top of other
rocks and the ones at the bottom will fail.

2) What is the shear strength of the mountain? No structure is
perfect and given enough shear, a fracture will cause the mountain to
break and the top to slide off. The wider the mountain the greater
the impact of shear, the thinner the lesser impact.

3) What is below the mountain? Make the mountain massive enough and
it will subside (sink) into the structure below. On Earth, we have
floating tectonic plates. Mountains are limited by this - not
coincidentally, it is believed that Mt. Everest sits atop two
overlapping plates so it doesn't sink into the liquid.

One of the equations I've seen that attempts to solve this problem is:

So, we can see that gravity is certainly a player, many mountains will
be limited by their components.

This answer would seem to confirm Jonas's explanation. However, it does not address some exogeological formations, such as Olympus Mons on Mars, which is approximately three times larger than similar formations found on Earth even though Martian gravity is about 53% that of Earth's. This seems to indicate that the relationship between gravity, geology, and maximal height may be too complex to be accurately summarized in a single rule of thumb or simple equation.

The original post on Quora can be found here, but a Quora account is necessary in order to view it.

Welcome to Skeptics! This is a pretty unreliable reference. Not peer-reviewed, no references, no authentication (is this person really an engineer from NASA?), no information about their area of expertise or qualifications that might make their claims-from-authority somewhat more trustworthy.
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Oddthinking♦Jul 8 '13 at 22:21

Answers here must be supported. The standard for publication isn't quite as high as an academic journal, but primary references (textbooks, professors' websites, NASA, journal articles that aren't behind a paywall) that are online help. Comments can have less proven material.
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PaulJul 9 '13 at 5:01

Though the above answer is not peer reviewed, identical information can easily be found in credible sources. For example: ias.ac.in/jarch/jaa/2/165-169.pdf In fact, this information is often preceded by "obviously...." "as is well known...." or "it is firmly established that...." Few of these academic sources, however, are very accessible to the layman, making the above answer an acceptable reduction.
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guestJul 9 '13 at 19:13